Precalculus Examples

$\log_{3} (\frac{5 r^{5}}{\sqrt[3]{t^{2}}})$

Step 1

Rewrite $\log_{3} (\frac{5 r^{5}}{\sqrt[3]{t^{2}}})$ as $\log_{3} (5 r^{5}) - \log_{3} (\sqrt[3]{t^{2}})$ .

$\log_{3} (5 r^{5}) - \log_{3} (\sqrt[3]{t^{2}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{t^{2}}$ as $t^{\frac{2}{3}}$ .

$\log_{3} (5 r^{5}) - \log_{3} (t^{\frac{2}{3}})$

Step 3

Expand $\log_{3} (t^{\frac{2}{3}})$ by moving $\frac{2}{3}$ outside the logarithm.

$\log_{3} (5 r^{5}) - (\frac{2}{3} \log_{3} (t))$

Step 4

Combine $\frac{2}{3}$ and $\log_{3} (t)$ .

$\log_{3} (5 r^{5}) - \frac{2 \log_{3} (t)}{3}$

Step 5

Rewrite $\log_{3} (5 r^{5})$ as $\log_{3} (5) + \log_{3} (r^{5})$ .

$\log_{3} (5) + \log_{3} (r^{5}) - \frac{2 \log_{3} (t)}{3}$

Step 6

Expand $\log_{3} (r^{5})$ by moving $5$ outside the logarithm.

$\log_{3} (5) + 5 \log_{3} (r) - \frac{2 \log_{3} (t)}{3}$